Electrical Power Systems

Power System Stability 279
Pi – Pe = M. d
2
qe dqe
2 + D. = Pa ...(11.13)
dt
dt
Where M has been defined in eqn. (11.8) or eqn. (11.9). D is a damping coefficient and q e
is the electrical angular position of the rotor. It is more convenient to measure the angular
position of the rotor with respect to a synchronously rotating frame of reference. Let
\
2
d = q e –w s .t ...(11.14)
2
d qe d d
= 2 2
dt dt
...(11.15)
Where d is the power angle of the synchronous machine. Neglecting damping (i.e., D = 0 )
and substituting eqn. (11.15) in eqn. (11.13), we get,
M. d
dt
Using eqns. (11.16) and (11.6), we get
2
d
= Pi – P
2
e MW ...(11.16)
2
d
= Pi – Pe MW ...(11.17)
GH d
p f 2
dt
Dividing throughout by G, the MVA rating of the machine,
M (pu) d
dt
where M(pu) = H
p f
2
d
=(P
2
i – Pe) pu ...(11.18)
2
...(11.19)
H d d
or
p f 2 = (Pi – Pe) pu ...(11.20)
dt
Eqn. (11.20) is called swing equation. It describes the rotor dynamics for a synchronous
machine. Although damping is ignored but it helps to stabilizer the system. Damping must be
considered in dynamic stability study.
11.3 MULTI-MACHINE SYSTEM
In a multi-machine system, a common system base must be selected. Let
Gmachine = machine rating (base)
Gsystem = system base.